Question

Magnetite, a magnetic oxide of iron used on recording tapes, crystallizes with iron atoms occupying 1/8 of the tetrahedral holes and ½ of the octahedral holes in a closed packed array of oxide ions. What is the formula of magnetite?

Answer 1

In a solid structure, the number of tetrahedral voids is 8 and the number of octahedral voids is exactly half of the tetrahedral voids, i.e., 4. So, by multiplying the given fraction of respective voids and then by taking the ratio we can calculate the formula.

Complete answer:
In a solid structure, the atoms are very close to each other, so there is very little space between the atoms, and these spaces are known as voids and there are two types of voids, i.e., octahedral voids and tetrahedral voids. In a crystal, the numbers of tetrahedral voids are always double the numbers of octahedral voids.
The given compound is magnetic oxide means the compound of iron and oxygen, as we know that the iron form cubic close packing and in ccp the number of tetrahedral voids is 2n and the numbers of octahedral voids are exactly half of the tetrahedral voids, i.e., n.
And given the iron atom crystallizes at 1/8 of tetrahedral voids and ½ of the octahedral voids. By keeping the ratio of iron and oxygen, we get:
Fe : O
$\text{2n x }\dfrac{1}{8}\text{ + n x }\dfrac{1}{2}\text{ : n}$
$\dfrac{3\text{n}}{4}\text{ : n}$
$3n\text{ : }4n$
Therefore, the formula will be $F{{e}_{3}}{{O}_{4}}$.
Hence the formula of magnetite is $F{{e}_{3}}{{O}_{4}}$.

Note:
A tetrahedral void is surrounded by 4 atoms, therefore, its coordination number is 4 while an octahedral void is surrounded by 6 atoms, therefore, its coordination number is 6.